Large deviation estimates of the crossing probability for pinned Gaussian processes

Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related t…

Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds

In this paper we study a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having a mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, whic…

Precise large deviations of the first passage time

Let $S_n$ be partial sums of an i.i.d. sequence $\{X_i\}$. We assume that $\mathbb{E} X_1 <0$ and $\mathbb{P}[X_1>0]>0$. In this paper we study the first passage time $$ τ_u = \inf\{n:\; S_n > u\}. $$ The classical Cramér's estimate of the ruin probability says that $$ \mathbb{P}[τ_u<\infty] \sim C e^{-α_0 u}\quad \text{as } u\to \infty, $$ for some parameter $α_0$. The aim of the paper is to describe precise large deviations of the first crossing by $S_n$ a linear boundary, more precisely for a fixed parameter $ρ$ we study asymptotic behavior of $\mathbb{P}\left[τ_u = \lfloor u/ρ\rfloor \righ…

利用 Girsanov 定理得到的边界穿越概率非显式公式

Non-explicit formula of boundary crossing probabilities by the Girsanov theorem

本文研究带随机漂移、随机方差的 Wiener 过程穿越单侧随机边界的概率，借助 Girsanov 定理与 Radon–Nikodym 导数分解给出非显式及更显式公式。

分数布朗运动任意Hurst指数的截断变化集中性，其1/H变化和局部时间

Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their $1/H$-variations and local times

本文研究了分数布朗运动任意Hurst指数的截断变化集中性，推导了偏离概率界，并建立了轨迹局部时间的弱收敛性。

首达渗流中随机度量在速度$n^d$下的大偏差原理

Large deviation principle at speed $n^d$ for the random metric in first-passage percolation

该研究针对$\u2124^d$上的标准首达渗流模型，通过多维次可加性论证等方法，证明了紧集上重整化首达时间满足速度为$n^d$的大偏差原理，并推导了相关推论。

Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-θ(d)} where θ(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(θ(d) + θ(2))}. For d=1, this connection allows for a physical realiz…

Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution

A conditioned stochastic process can display a very different behavior from the unconditioned process. In particular, a conditioned process can exhibit non-Gaussian fluctuations even if the unconditioned process is Gaussian. In this work, we revisit the Ferrari-Spohn model of a Brownian bridge conditioned to avoid a moving wall, which pushes the system into a large-deviation regime. We extend this model to an arbitrary number $N$ of non-crossing Brownian bridges. We obtain the joint distribution of the distances of the Brownian particles from the wall at an intermediate time in the form of the…

泊松计数过程与带有偏移的线性移动边界的第一通过问题

A first passage problem for a Poisson counting process with a linear moving boundary

本文研究泊松计数过程与线性移动边界的第一通过时间问题，通过直接时域方法和拉普拉斯域方法统一分析，推导出亚临界区域的大偏差函数和任意偏移下的条件均值第一通过时间的闭式表达式。

矩阵值随机Schrödinger算子谱行列式的大偏差与立方势中的Dyson Brownian motion

Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials

本文研究矩阵值随机Schrödinger算子谱行列式的大偏差，通过映射到立方势中的Dyson Brownian motion，计算了有限N下的越障概率，给出了谱边缘下方平均态密度的指数尾部估计，并独立推导了无序弹性弦驻点的总复杂度。